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Semiclassical and variational approaches to vison
dispersion and condensation
Grégoire Misguich, Frédéric Mila
To cite this version:
Grégoire Misguich, Frédéric Mila. Quantum dimer model on the triangular lattice: Semiclassical and
variational approaches to vison dispersion and condensation. Physical Review B: Condensed Matter
and Materials Physics (1998-2015), American Physical Society, 2008, 77, pp.134421.
�10.1103/phys-revb.77.134421�. �cea-02014494�
Quantum dimer model on the triangular lattice: Semiclassical and variational approaches
to vison dispersion and condensation
Grégoire Misguich*
Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Frédéric Mila
Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland 共Received 7 December 2007; revised manuscript received 23 January 2008; published 9 April 2008兲
After reviewing the concept of vison excitations inZ2dimer liquids, we study the liquid-crystal transition of
the quantum dimer model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a “soft-dimer” version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and
冑
12⫻冑
12 symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the Rokhsar-Kivelson共RK兲 point, which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave functions at the RK point.DOI:10.1103/PhysRevB.77.134421 PACS number共s兲: 05.50.⫹q, 71.10.⫺w, 75.10.Jm
I. INTRODUCTION
Since they have been shown to possess resonating valence bond 共RVB兲 phases on the triangular,1 kagome,2 and other
共nonbipartite兲 lattices,3quantum dimer models共QDMs兲 have
been one of the main paradigms in the field of quantum spin liquids. These models, where the Hilbert space is spanned by hard-core dimer coverings of the lattice, are expected to cap-ture the phenomenology of quantum antiferromagnets where the wave function is dominated by short-range valence bond configurations. On the triangular lattice, the simplest QDM is defined by the Hamiltonian:
H = −t r + H.c. +V r + 共1兲 where the sum runs over all plaquettes 共rhombi兲 including the three possible orientations. The kinetic term, controlled by t, flips the two dimers on every flippable plaquette, i.e., on every plaquette with two parallel dimers, while the potential term controlled by the interaction V describes a repulsion 共V⬎0兲 or an attraction 共V⬍0兲 between nearest-neighbor dimers.
The RVB phase is now relatively well understood. The first result goes back to Rokhsar and Kivelson,4who showed
that, for V/t=1 共the Rokhsar-Kivelson or RK point兲, the ground state is the sum of all configurations with equal am-plitudes:
兩RK典 =
冑N
1兺
c
兩c典. 共2兲
Since then, dimer-dimer correlations have been shown to be short ranged in a range of parameters below V/t=1,1,5 and
the excitation spectrum to be gapped.1,6–8Besides, it has
to-pologically degenerate ground states on non-simply-connected clusters.1,5,7 This degeneracy is not related to a
standard symmetry breaking. Indeed, there is no local order parameter,5,9but onlyZ
2 topological order.10
In QDM, the nature of the phase transition from a liquid to a solid is a long standing problem. It goes back to Jalabert and Sachdev11 共see also Ref. 12兲, who studied a
three-dimensional frustrated Ising model related to the square lat-tice QDM.
In a more general context, Senthil and Fisher13,14showed that models of Mott insulators can be cast in the form of aZ2
gauge theory. In this language, the transitions from a frac-tionalized insulator to a conventionally ordered insulator ap-pears to be a condensation of Z2 vortices 共dubbed visons兲.
Using a duality relation,15 they showed that such transitions
correspond to an ordering transition in a frustrated Ising model in transverse field. As we will see, this applies to the present QDM.
Building on a mapping between QDM’s at V = 0 and Ising models in a transverse field, Moessner et al.16,17 have
devel-oped a Landau-Ginzburg approach and suggested that the transition could be continuous and in the three-dimensional O共4兲 universality class.
Numerical evidence in favor of this scenario has been obtained with Green’s function quantum Monte Carlo by Ralko et al., who have shown that, at the transition point, the static form factor of the crystal decreases to zero on the crystal side of the transition, while the dimer gap also de-creases to zero on the liquid side.18More recently, the vison spectrum has also been numerically determined using Green’s function quantum Monte Carlo, with the conclusion that a soft mode, indeed, develops at the transition.19
In spite of these results, a simple picture for the wave functions of the visons and the evolution of their spectrum is still missing. One difficulty is that, like any vortex, these excitations cannot be created by operators which are local共in the dimer variables兲. To attack this problem, we follow two strategies. First of all, we look at the problem in the context of aZ2gauge theory on the triangular lattice. As usual, this
theory can be mapped onto a dual Ising model.15The duality
transforms the nonlocal excitations of the gauge theory into local excitations, which we study using a semiclassical ap-proach. This simple 1/S expansion already captures most aspects of the confinement-deconfinement transition 共soft mode and condensation兲 of the Z2 gauge theory. Second,
building on the explicit form of the excitations of the Z2
gauge theory, we construct single vison wave functions for the QDM and study the properties of the system at the RK point in the variational Hilbert space spanned by these states. These “variational visons”—living on the triangular plaquettes and experiencing a flux emanating from each site of the lattice—turn out to be linearly independent. The associated dispersion is in good qualitative agreement with that obtained in Monte Carlo simulations, and the value of the gap at the RK point共⌬var= 0.119兲, obtained with a single
variational parameter, has the correct order of magnitude 共Monte Carlo simulations6give 0.089兲. Improving
quantita-tively further these variational results would require more adjustable parameters to account in more details for the local 共dimer-dimer, etc.兲 correlations in the vicinity of the core of the vortex, a task which has not been carried out here.
II. VISONS INZ2GAUGE THEORY
The connection between QDMs andZ2gauge theories has
already been discussed from different perspectives 共see, in particular, Refs.2,3,12, and 20兲, and is rooted in the
exis-tence, in both families of models, of Ising-like degrees of freedom subjected to local constraints共hard-core constraints for the dimers, Gauss law for the gauge theory兲.
In this section, we review two known mappings:共i兲 from aZ2 lattice gauge theory on the hexagonal lattice to the
tri-angular lattice QDM 共valid in a particular limit, Sec. II A兲 and共ii兲 from the Z2theory to its dual frustrated Ising model
共Sec. II B兲. The latter Ising model is then studied using a semiclassical approximation in Secs. II B 2 and II B 3. Since theZ2gauge theory-QDM mapping is formally justified only
in the confined phase of the gauge theory, the relevance of the results obtained in the RVB 共deconfined兲 phase of the QDM is not guaranteed a priori. Some reasons why this is expected to be the case are discussed in the next section, when we build on the results obtained for the excitations of theZ2 gauge theory in its deconfined phase to construct
el-ementary excitations in the RVB phase of the QDM.
A.Z2gauge theory
To write down aZ2 gauge theory analogous to a QDM,
the starting point is to define Pauli matricesជlon the bonds l
of the triangular lattice such thatl x
= −1 if bond l is occupied by a dimer, and +1 if it is empty, whilelzchanges the state of bond l. Since each kinetic term of the QDM flips the dimers around a rhombus, it corresponds to a product of four
l z
around this rhombus. This would, however, not keep the structure of the simple Z2 gauge theories, in which the
ki-netic term acts on an elementary plaquette of the lattice, a crucial ingredient to get a simple Ising model by duality.15
An alternative is to consider the more standard Z2 gauge
theory which, in its Hamiltonian formulation, is defined by
H = HJ+ H⌫= − J
兺
l l x −⌫兺
i兿
l共i兲 l共i兲 z , 共3兲where i runs over the sites of the dual honeycomb lattice, and l共i兲 are the three bonds forming the triangular plaquette around site i 共see Fig. 1兲. The hallmark of this model is to
have local conserved quantities. Indeed,
冋
H,兿
l关a兴 l关a兴
x
册
= 0, 共4兲
where a is a site of the triangular lattice, and the product over l关a兴 runs over the six links emanating from a 共see Fig. 1兲.
This allows one to define different sectors according to whether兿l关a兴关a兴x is equal to +1 or −1. Since in the QDM the
number of dimers emanating from a given site is exactly equal to 1, it is clearly better to consider the sector where
兿
l关a兴 l关a兴
x
= − 1 共5兲
for all a since this forces the number of dimers emanating from a site to be odd共defining an odd Ising gauge theory in the terminology of Ref. 20兲. Then, the true constraint is
re-covered in the limit ⌫/J→0 if J⬎0 since, in the ground state, the number of dimers is then minimal. A bona fide QDM is then recovered if H⌫is treated with degenerate per-turbation theory. Since H⌫changes the number of dimers, its effect vanishes to first order. To second order, however, one recovers exactly the QDM of Eq. 共1兲 with V=0 and t
=⌫2/J. So, if J/⌫Ⰷ1, the Z2 gauge theory maps onto the
QDM at V/t=0. At finite J/⌫, the model can be viewed as a “soft-dimer” model, where one, three, or five dimers may touch a given site.
As we shall see below, the limit J/⌫Ⰷ1 lies deep inside the confined phase of the gauge theory, and from previous work on the QDM, it is known that at V/t=0, the model is in a valence bond crystal phase. It would, of course, be very interesting to connect the two models away from this limit, when the gauge theory is in its deconfined phase and the QDM is in the dimer liquid phase. An interesting step in this
a
1)
2)
l[a]
3)
l<i
l(i)
i
i
FIG. 1. 共Color online兲 Some useful definitions of sets of bonds: 共1兲 l共i兲 is the set of the three bonds forming the edges of the plaquette i.共2兲 l关a兴 is the set of the six 共fat兲 bonds emanating from the triangular site a.共3兲 l⬍i denotes the bonds forming a “zigzag” string extending共to the left兲 from the triangular plaquette i to an edge of the lattice.
direction is provided by higher order perturbation theory. In-deed, to fourth order in⌫/J, a repulsion between dimers V =⌫4/2J3 is generated. However, other terms of the same
or-der, involving dimer shifts along loops of length 6, are also generated 共see Appendix A兲. So, a rigorous mapping does not extend beyond the V/t=0 point.
B. Dual Ising model
As usual, thisZ2 gauge theory is best analyzed by
map-ping it onto a dual Ising model in a transverse field.
1. Model
This can be achieved by introducing spin-21 operatorsជi
on the dual honeycomb lattice:
i x =
兿
l共i兲 l共i兲 z , i z =兿
l⬍i l x , 共6兲where l⬍i represents all bonds cutting a straight path 共say, horizontal, see Fig.2or1兲 starting at i 共we implicitly assume
a finite lattice with open boundary conditions兲. Combined with the constraint, this definition implies that
i z j z = Mijl x 共7兲
for two neighboring sites i , j separated by the bond l. Mij
=⫾1 is such that each hexagon has exactly one Mij= −1
bond 共see Fig. 3兲. In terms of these spin operators, the
Hamiltonian is the fully frustrated Ising model21共FFIM兲
dis-cussed by Moessner and Sondhi:17,26
H = HJ+ H⌫= − J
兺
具i,j典 Miji z j z −⌫兺
i i x . 共8兲Note that choosing other paths to define i z
leads to other signs for Mij, which, however, are always such that an odd
number of minus signs appear around each hexagon, leading to the same Ising model up to a gauge transformation. The present choice leads to the smallest unit cell 共four honey-comb sites兲.
In the following, we study the phase diagram and the excitations of this model within a semiclassical共large S兲 ap-proximation.
2. Classical phase diagram
First, we determine the classical ground state of the model as a function of ⌫/J. To this end, we replace the spin-12 operators by classical three-component vectors of unit length. Since the y component does not appear in the Hamil-tonian, it is clear that in the ground state, the magnetization must lie in the x-z plane. To find the lowest energy solution, we use the following parametrization:
i
z→共i兲, 共9兲 i
x→
冑
1 −共i兲2, 共10兲 with兩共i兲兩艋1. The corresponding energy is
E = −J 2
兺
ijMij共i兲共j兲 − ⌫
兺
i冑
1 −共i兲2. 共11兲Initially defined for nearest neighbors关Eq. 共7兲兴, the
coef-ficients Mijhave been upgraded to a matrix M by setting all
other elements to 0. This matrix describes the motion of a particle on a honeycomb lattice with four sites per unit cell and a flux per hexagonal plaquette. It reduces to a 4⫻4 matrix after Fourier transformation. The eigenvalues associ-ated with the momentum k =共kx, ky兲 are17
mk
1,2,3,4
= ⫾
冑
3⫾冑
2关3 + cos共2kx兲 − cos共kx+ ky兲 + cos共kx− ky兲兴.Likewise, we denote by the column vector of components 兵i其.
⌫/J⬎
冑
6. For ⌫ⰇJ, we expect the共i兲’s to be small 共or zero兲. We can, therefore, expand Eq. 共11兲 to quadratic order:E = −J 2 t M+1 2⌫ t . 共12兲
The largest eigenvalue of M being
冑
6, we find that, as long as ⌫/J⬎冑
6, the energy is minimized by= 0 and all spins point in the x direction.0⬍⌫/J艋
冑
6. At⌫/J=冑
6, all the real eigenvectors of M for the eigenvalue冑
6 共satisfying 兩i兩艋1兲 minimize Eq. 共12兲.ij
Ω
j
i
FIG. 2. 共Color online兲 Dashed lines: Horizontal paths used in Eq.共6兲 connecting the boundary of the system to triangles i and j.
Dotted segment: Path⍀ij. In this example,⍀ijcrosses one dimer of the reference configuration c0共ellipses兲 and, therefore,⑀ij= −1关Eq.
共47兲兴.
4
1 2
3
y
x
FIG. 3. 共Color online兲 The fat 共green兲 bonds of the hexagonal lattice 共crossed by a dimer of the reference configuration on the triangular lattice兲 have Mij= −1, the other bonds have Mij= 1. The
four sites of the unit cell共small blue triangles兲 are labeled from 1 to 4, and the Bravais vectors are x and y.
Let us choose four complex vectors which form a basis of the subspace associated with the eigenvalue
冑
6:共Ref. 17兲v1共x,y兲 = v3*=
冤
e5i/12/F e−i/6/F 1 e−i/12冥
exp冉
i 6 x + i 2y冊
, v2共x,y兲 = v4*=冤
ei/12 e−5i/6 1/F e−5i/12/F冥
exp冉
i5 6 x + i 2y冊
, F = 2 sin共5/12兲, 共13兲 where the four entries of the vector refer to the four sites in the unit cell共numbered as in Fig.3兲 and x,y are the 共Bravais兲coordinates of the unit cell. These eigenvectors correspond to the four points labeled B in the rectangular Brillouin zone of Fig.6.
The most general real eigenvector can be parametrized by three angles␣1,␣2, and and a normalization factor:
共i兲 = 关cos共兲R„v1共i兲ei␣1… + sin共兲R„v2共i兲ei␣2…兴.
共14兲 To find the ground state, the energy has to be minimized as a function of these three parameters. The analysis is made easier when one realizes that the second and fourth moments of the spin deviations are independent of the three angles. Indeed, 1 N
兺
i 共i兲2= r22, 共15兲 1 N兺
i 共i兲4= 2r44, 共16兲 with r2=1 4冉
1 + 1 F2冊
. 共17兲Replacingby Eq.共14兲 in the expression for the energy 关Eq.
共11兲兴, we get up to order 4: E N= −⌫ + 1 2共⌫ − J
冑
6兲r 22+1 4⌫r 44+O共6兲. 共18兲Minimizing with respect to2 gives 2=J
冑
6 −⌫f2⌫ . 共19兲
For⌫⬎J
冑
6, the energy is minimized for=0. For ⌫艋J冑
6, we have to expand Eq. 共11兲 to the sixth order to find theangles ␣1, ␣2, and  which minimize the classical energy.
Indeed, Eq. 共18兲 shows that, up to order 4, the energy is
independent of the three angles. At this order, the energy is constant and minimum on a three-dimensional sphere, a
con-sequence of the O共4兲 symmetry discovered by Moessner and Sondhi.17One, therefore, has to go to the next order to find
the actual minima. At sixth order in , the energy is mini-mized when
1 N
兺
i共i兲6 共20兲
is minimum. A numerical investigation shows that the solu-tions共for ⌫/J close to but below
冑
6兲 are 48-fold degenerate and can be deduced from each other by symmetry operations 共12 translations and 4 point-group operations兲. This confirms the result obtained previously on symmetry grounds.16,17Motivated by the FFIM–dimer model correspondence, we are interested in the average “dimer density:”
dij= 1 2共1 − Mij具i z j z典兲 共21兲
for all pairs共i, j兲 of nearest-neighbor honeycomb sites. In the classical limit, this may be approximated by
dij=
1
2关1 − Mij共i兲共j兲兴, 共22兲 where 关Eq. 共14兲兴 is the classical ground state. Close to the
transition at⌫c= J
冑
6, scales as⬃冑
⌫c−⌫ 关Eq. 共19兲兴 andthe dimer density, thus, shows small deviations about 12. To visualize the “dimerization” pattern in the vicinity of⌫c, the
appropriate quantity is, therefore, a relative rescaled “dimer” density defined by Dij=
1 2共dij−
1
2兲, and plotted in Fig.4 for
one of the 48 ground states. The obtained pattern is highly reminiscent of the
冑
12⫻冑
12 valence bond crystal 共VBC兲 observed in the triangular lattice QDM. In particular, the 48 spin configurations give only 12 different dimer patterns, with a smaller unit cell containing 12 sites of the triangular lattice. It is also interesting to notice that共i兲 vanishes in the triangles located inside the large diamonds 共marked with aFIG. 4.共Color online兲 Plot of the rescaled relative dimer density Dijin one of the 48 classical ground states of the FFIM in
trans-verse field, for a transtrans-verse field⌫ just below ⌫c. Thin red bonds represent Dij= 0 共corresponding to the highest dimer density dij
= 0.5兲. Blue bonds are for Dij⬍0, that is, a lower dimer density,
with a width proportional to兩Dij兩. Dijtakes only four different val-ues: 0, −0.259, −0.518, and −0.776.
dot in Fig.4兲. From Eq. 共10兲, the corresponding sites have a
magnetization pointing exactly in the x direction, which means a total absence of vison and local correlations identi-cal to that of the J = 0 paramagnetic ground state. The dimer density dij is uniform inside these diamonds, in qualitative
agreement with the intriguing observation made in Ref.18.
3. Semiclassical analysis for⌫ÕJÐ
冑
6If J = 0, the ground state is the fully polarized state i x
= 1. The first共degenerate兲 excited state is obtained by flip-ping one spin at some arbitrary triangle. To first order in J/⌫, the spectrum is obtained by diagonalizing the perturbation HJin this subspace. This amounts to diagonalizing M, and
leads to the eigenvalues ⑀ki= 2⌫+Jmki, a result already ob-tained previously.17
To go further, beyond the limit J/⌫Ⰶ1, we perform a 1/S semiclassical expansion. We generalize the spin-1/2 Hamil-tonian to an arbitrary value S of the spin:
H = − J S2具i,j典
兺
MijSi z Sj z −⌫ S兺
i Si x 共23兲关which reduces to Eq. 共8兲 when S=1/2兴. Spin deviations
away from the x directions are represented using Holstein-Primakoff bosons. To leading order in 1/S:
Si z =1 2
冑
2S共bi † + bi兲, 共24兲 Si x = S − bi†bi. 共25兲One truncates the Hamiltonian to quadratic order in bi, and
diagonalizes it through a Bogoliubov transformation: bi†=
兺
j
Uija†j+
兺
jVijaj. 共26兲
For the a†j operators to be bosonic creation operators, the matrices U and V must satisfy U†U − V†V = 1. It is convenient to introduce a unitary matrix⍀ which transforms M into a diagonal matrix M˜ :
M =⍀M˜ ⍀†, 共27兲
M˜kp= mk␦kp. 共28兲
In this new basis, we can look for diagonal solutions for U and V:
U =⍀U˜ ⍀†, V =⍀V˜⍀†, 共29兲 U
˜
kp= uk␦kp, V˜kp=vk␦kp. 共30兲
Since M is real, we can choose⍀苸O共N兲. After some alge-bra, one finds that the uk and vk which diagonalize H are
given by uk= cosh共k兲, 共31兲 vk= sinh共k兲, 共32兲 tanh共2k兲 = − mk mk+ 2⌫/J . 共33兲
The energies of the Bogoliubov excitations are given by
⑀k= J S
冑
⌫ J冉
mk+ ⌫ J冊
. 共34兲As expected, the spectrum becomes gapless at⌫/J=
冑
6 and we recover the localized spin flip energy ⑀k⯝⌫/S when ⌫→⬁. The Bogoliubov spectrum is shown Fig. 5 for a few values of⌫ at or above ⌫c. One clearly sees that the
disper-sion is linear around the gapless points at the transition. This behavior is remarkably similar to that found numeri-cally by Ralko et al.19 for the QDM by Green’s function
quantum Monte Carlo. In the next section, we build on this resemblance to develop a variational approach to the vison spectrum of the QDM. Note that, as stated above, the Z2
gauge theory and the QDM can only be rigorously mapped onto each other deep into the confined共VBC兲 phase, and the resemblance between their spectra in the deconfined共RVB兲 phases might seem fortuitous. A somewhat deeper connec-tion will be described in the next secconnec-tion.
B B B B 000 0 444 4 A C A C A C A C 333 3
ε(κ)
ε(κ)
ε(κ)
ε(κ)
111 1 A A A A 222 2FIG. 5. 共Color online兲 Spin-wave dispersion relation S⑀k/J 关Eq. 共34兲兴 for the path A→B→C→A in the Brillouin zone 共see Fig.6兲
and different values of⌫/J: 3 共top兲, 2.75, 2.5, and
冑
6⯝2.448 共bot-tom兲. At the critical point ⌫/J=冑
6, the spectrum is linear around k =共/6,/2兲 and k=共5/6,/2兲.A
B
C
kx kyFIG. 6. 共Color online兲 Dashed rectangle: Brillouin zone 共− 艋kx艋 and −艋ky艋兲 of the hexagonal lattice shown in Fig.3.
A, B, and C are the high-symmetry points used in Fig.5. The large hexagon is the Brillouin zone of the underlying triangular lattice.
III. VISONS IN THE QUANTUM DIMER MODEL
In the Ising model, elementary excitations for large enough⌫/J are spin flips, induced by i
z
, which delocalize and get dressed under the effect of HJ. In the equivalent dual
gauge theory, this excitation is produced by the nonlocal string operatori
z
=兿l⬍il x
. By analogy, it is natural to define a “point” vison creation operator for the QDM by5,22
Vi=共− 1兲Nˆ 共i兲, Nˆ 共i兲 =
兺
l⬍inˆl, 共35兲
where the dimer operator nˆlis defined by nˆl= 1 if bond l is
occupied, and 0 otherwise. Again, we consider here a finite lattice with open boundary conditions. Let us first see to which extent this operator is the analog of the vortex creation operator in Ising gauge theories.
A.Z2gauge structure of quantum dimer models In gauge theories, the Wilson loop operator defined by
W⍀=
兿
l苸⍀ l
z 共36兲
plays a central role since it allows one to distinguish共in the absence of matter field, as here兲 the deconfined and confined phases depending on whether its ground state expectation value共or flux兲 tends to zero exponentially with the perimeter of the domain⍀ 共deconfined兲 or with the area of the domain 共confined兲. For a Z2gauge theory, W⍀2 = 1 and the flux going
through⍀ measured by W⍀can only take two values⫾1. In that respect, an interesting property of the operator i z
=兿l⬍il x
is that it changes the flux between −1 and +1 if the site i is inside the domain. Then, i
z j
z
commutes with the Wilson loop and does not change the flux unless i and j sit on opposite sides of the boundary. Since the deconfined phase can be accessed from the JⰆ⌫ limit, the ground state can be thought of as an Ising paramagnet共x= 1 and W
⍀= 1
every-where兲 “dressed” perturbatively by successive applications of Ji
z j z
, where i and j are nearest neighbors. Since Ji z
j z
only changes the flux for pairs ij across the boundary, the expectation value of the Wilson loop operator behaves ac-cording to a perimeter law. A similar perturbative argument can be used to derive the area law in the confined phase.
Most of these standardZ2 gauge theory results apply to
the QDM, with one difficulty however. The Wilson loop op-erator cannot be defined in the same way since flipping the dimer occupation along a loop will often lead to an unphysi-cal state that violates the condition of having exactly one dimer emanating from each site. Different ways to overcome this problem can be envisaged. One possibility is to accept that the Wilson loop operator gives zero when applied to states that are nonflippable along the loop. The property W⍀2 = 1 is lost, but W⍀ has eigenvalues 0, +1, or −1, and eigenstates with eigenvalues +1 or −1 are still interchanged when a vison operator 关Eq. 共35兲兴 is applied inside the
do-main. So confinement or deconfinement of visons is still ex-pected to lead to area or perimeter laws.
Alternatively, starting from the observation that, when it does not give zero, the Wilson loop operator just shifts the
dimers along the contour, one can try to define a 共compli-cated兲 flux operator in the dimer space which shifts the dimer along a “fattened” contour which “adapts” locally to the dimer configuration it acts upon. Although tricky to define in practice,27 this operator would have the advantage that its
square is still equal to 1, preserving the manifestZ2structure
of the theory.
Another way to underline the deep connection between the two models is to consider the Ising model as a soft-dimer model and introduce the projection operator onto the hard-core dimer Hilbert space, defined by
Pˆ =
兿
p
共nˆp− 3兲共nˆp− 5兲
共1 − 3兲共1 − 5兲 , 共37兲 where the operator nˆpcounts the number of frustrated bonds
共dimers兲 around the plaquette p: nˆp= 1 2
兺
i=1 6 共1 − Mi,i+1i z i+1 z 兲. 共38兲By construction, Pˆ =1 on all the Ising configurations, which correspond to a valid hard-core dimer covering共of the trian-gular lattice兲 and Pˆ=0 otherwise. Now, Pˆ commutes with 兿ii
x
since all terms in Pˆ contain an even number of z
operators. This means that Pˆ conserves the total flux, so that a spin state with −1 flux becomes a dimer wave-function with an odd number of visons after projection.
B. Point vison
Equation 共35兲 defines the simplest operator which
changes the flux, that is, which creates a vison. So we may consider 兩i典 = V共i兲兩RK典 =
冑N
1兺
c 共− 1兲Nˆ 共i兲兩c典 共39a兲 =冑N
1 Pˆi z兩兵 j x= + 1其典 共39b兲as a first variational approximation to the true lowest eigen-state of H in the −1 flux sector 共N is the total number of dimer coverings兲. The ground state 兩RK典 is a zero-energy eigenstate ofH, implying that the expectation value EK⬍0
of the kinetic energy term exactly compensates the expecta-tion value EP⬎0 of the potential energy term. Indeed,
EP= 3Np0= − EK, 共40兲
where 3N is the total number of rhombi共N the number of sites兲 and p0the probability to have two parallel dimers on a
given rhombus in the classical dimer problem共with uniform measure over all dimer configurations兲. Using the Pfaffians method,23,24 one finds in the thermodynamic limit
In兩i典 the expectation value of the potential energy term is the same as in兩RK典. In fact, all the observables which are diagonal in the dimer basis commute withzand, thus, have the same expectation value in the ground state 兩RK典 and in the trial wave function兩i典. The increase of the energy is only kinetic. Because the kinetic terms corresponding to the three diamonds around i anticommute with i
z
, their expectation values have changed sign. Thus, we find
具i兩H兩i典 − 具RK兩H兩RK典 = 6p0⯝ 0.56. 共42兲
C. Dispersing point vison
One can lower the energy by constructing plane waves. To compute the associated dispersion relation we have to evalu-ate the following matrix elements:
Sij0=具i兩j典, 共43兲
Hij0=具i兩H兩j典. 共44兲
We begin with the overlap matrix Sij
0
= 1 N
兺
c具c兩共− 1兲Nˆ 共i兲+Nˆ共j兲兩c典 共45兲
关Nˆ共i兲 and Nˆ共j兲 are defined in Eq. 共35兲兴 which simplifies to
Sij0=⑀ij
1 N
兺
c具c兩共− 1兲Nˆ 共i,j兲兩c典, 共46兲 ⑀ij=共− 1兲N0共i,j兲, 共47兲
where the local operator Nˆ 共i, j兲 counts the number of dimers across some path ⍀ij connecting the triangles i and j
共see Fig.2兲, and N0共i, j兲 is equal to that number in the
refer-ence configuration c0, chosen with all the dimers horizontal 共Fig. 2兲. This follows from two simple properties:
具c兩共−1兲Nˆ 共i兲+Nˆ共j兲+Nˆ共i,j兲兩c典 is independent of the configuration c,
and具c0兩共−1兲Nˆ 共i兲+Nˆ共j兲兩c0典=1. We finally write
Sij
0
=⑀ij具共− 1兲Nˆ 共i,j兲典, 共48兲
where具¯典 represents the average with equal weight over all dimer coverings.
The average of any such diagonal observable can be com-puted using the Pfaffian of a 共modified兲 Kasteleyn matrix.23,24 In the present case, the “string” observable
共−1兲Nˆ 共i,j兲 is coded in the Kasteleyn matrix by changing the
signs of the matrix elements corresponding to bonds crossed by⍀ij, i.e., setting some bond fugacities to −1. To evaluate
numerically such an expectation value, we construct the modified Kasteleyn matrices corresponding to a large enough triangular lattice, in which the path⍀ijis embedded.
Finite-size effects decay exponentially with the system Finite-size, so that lattices with 28⫻28 sites 共with periodic boundary condi-tions兲 can safely be used to evaluate Sij0 with high accuracy
up to distances d⯝10 between triangles i and j. The matrix elements Sij
0
, plotted in Fig.7as a function of
the distance between i and j, decay exponentially 共a result anticipated by Read and Chakraborty22兲. These quantities
have already been evaluated by Ioselevich et al.5 using a classical Monte Carlo sampling.
By construction, a product like ⑀i1,i2⑀i2,i3¯⑀in,i1 共where
i1, i2, . . . , in form a closed loop of triangles兲 is equal to the
parity of the number of sites enclosed in the loop. Thus, the signs of the matrix elements Sij0 are similar to those of the hopping amplitude of a particle moving on the hexagonal lattice and subjected to a magnetic field corresponding to half a flux quantum per hexagon.28 In such a case, the
mag-netic unit cell has to be doubled compared to the original lattice cell. Since the original unit cell contains one triangu-lar site and two triangutriangu-lar plaquettes, the magnetic one con-tains four triangular plaquettes and, thus, four sites of the hexagonal lattice. In Fourier space, S0共k兲 is a 4⫻4 matrix, as
the matrix M discussed previously. The same is also true for the Hamiltonian matrix elements described below.
The spectrum of the overlap matrix S0共k兲 is plotted in Fig.
8 共for k describing a representative path in the Brillouin
zone兲. As an important result, the eigenvalues are strictly positive for all k.
Let us now turn to the matrix elements of the Hamil-tonian. We wish to transform Eq. 共44兲 into an expression
which can be evaluated using the Pfaffians, that is, the ex-pectation value of a diagonal observable in the dimer basis. First, we writeH as a sum of projectors
H = 2
兺
r0
⌸ˆr0, 共49兲
FIG. 7. 共Color online兲 Overlap 兩具i兩 j典兩 between two point-vison states关defined in Eq. 共39a兲兴 as a function the distance dijbetween i and j. The共blue兲 squares correspond to Eq. 共43兲 and the three 共red兲
crosses correspond to Eq.共59兲 with␣=−0.8. The calculations are
done using an exact evaluation of the Pfaffians on a 28-site lattice with periodic boundary conditions. Dashed line: guide for the eye corresponding to an exponential decay with the dimer-dimer corre-lation length−1= 0.76共Ref.5兲.
⌸ˆr0=兩r0典具r0兩, 共50兲
|ψ
r0=
1
√
2
−
共51兲and expand Eq.共44兲 into
Hij 0 = 2 Nc
兺
1,c2兺
r 0 共− 1兲N共c1,i兲+N共c2,j兲具c 1兩⌸ˆr0兩c2典, 共52兲with the notation N共c,i兲=具c兩Nˆ共i兲兩c典= ⫾1. Because of the double sum兺c1,c2, this does not yet have the form of a
diag-onal observable amenable to an evaluation with Pfaffians. To go further, we note that具c1兩⌸ˆr0兩c2典 vanishes if c1and/or c2is
not flippable around the rhombus r0. In addition,
具c1兩⌸ˆr0兩c2典=0 if c1and c2differ anywhere outside r0. So we
can restrict the sum to pairs of configurations 共c,c¯兲, which differ by a dimer flip in r0and which are identical elsewhere
on the lattice: Hij0= 2 N
兺
r0 共c,c¯兲兺
Fr 0共c兲=1 关共− 1兲N共c,i兲具c兩 + 共− 1兲N共c¯,i兲具c¯兩兴⌸ˆ r0关共− 1兲N共c,j兲 ⫻兩c典 + 共− 1兲N共c¯,j兲兩c¯典兴, 共53兲where Fr0共c兲=1 if c is flippable on rhombus r0, and Fr0共c兲
= 0 otherwise.
c and c¯ only differ inside r0, so the signs 共−1兲N共c,i兲 and
共−1兲N共c¯,i兲
are the same if i is not inside r0, and are opposite if
i苸r0. Let us note ␦i,r0= −1 if i苸r0, and ␦i,r0= 1 otherwise.
This leads to Hij 0 = 2 N
兺
r0 共c,c¯兲兺
Fr 0共c兲=1 共− 1兲N共c,i兲+N共c,j兲关具c兩 +␦ i,r0具c¯兩兴⌸ˆr0关兩c典 +␦j,r 0兩c¯典兴. 共54兲Using the explicit form of⌸ˆr0, we get
关具c兩 +␦i,r0具c¯兩兴⌸ˆr0关兩c典 +␦j,r0兩c¯典兴 = 1 2共1 −␦i,r0兲共1 −␦j,r0兲 共55兲 and, finally, Hij 0 =
兺
r0 共1 −␦i,r0兲共1 −␦j,r0兲 1 2N兺
c Fr 0共c兲=1 共− 1兲N共c,i兲+N共c,j兲 共56a兲 =⑀ij兺
r0 共1 −␦i,r0兲共1 −␦j,r0兲 1 2N兺
c Fr 0共c兲=1 具c兩共− 1兲Nˆ 共i,j兲兩c典. 共56b兲 The last expression is the average of a diagonal observable and can, thus, be evaluated using Pfaffians:Hij0= 1 2⑀ij
兺
r0 共1 −␦i,r0兲共1 −␦j,r0兲具共− 1兲 Nˆ 共i,j兲Fˆ r0典, 共57兲where we used an operator notation for the “flippability” Fˆr0兩c典=Fr0共c兲兩c典. To get a nonzero contribution, there must
be at least one rhombus r0 containing i and j. So Hij0= 0 if i and j are not nearest neighbors. Using modified Kasteleyn matrices in a way similar to that described for the overlap matrix S0, the nonzero matrix elements can be calculated. We
find兩Hij0兩=6p0for i = j;兩Hij0兩=2p0when i and j are first neigh-bors.
Although the matrix elements ofH are simple, the band structure is, however, not that of a simple tight-binding Hamiltonian, because of the nonorthogonality of the present variational vison states. In Fourier space, S0共k兲 and H0共k兲 are
4⫻4 matrices. The spectrum is obtained by solving the gen-eralized eigenvalue problem H0共k兲k− E共k兲S0共k兲k= 0. The
results are shown in Fig.9共␣= 0 curve兲. The minimum of⑀k
is found at共kx, ky兲=共/6,/2兲, in agreement with the Monte
FIG. 8. 共Color online兲 Spectrum of the overlap matrix S␣共k兲 关Eqs. 共43兲 and 共59兲兴, for different values of ␣ 共0, −0.2, −0.5, and
−0.8兲, along the path A→B→C→A in the Brillouin zone 共see Fig.
6兲. The bottom panel is a zoom on the lowest eigenvalues of S␣. In these calculations, the matrix elements Sij␣ are neglected for tri-angles i and j at distance d艌dmax= 10共102th neighbor on the hex-agonal lattice兲. Up to this distance, the finite-size lattice used in the calculation共28⫻28 sites兲 gives practically the infinite volume limit for Sij␣. The value of dmaxused here is large enough to ensure a good
convergence of the spectrum since the curves obtained with a smaller truncation distance共dmax⯝8.47—shown here with the same colors兲 are almost superposed with that for dmax= 10, except for the
bottom of the spectrum at␣=−0.5 and ␣=−0.8. The overlap spec-trum turns out to be gapped for␣=0 and ␣=−0.2 共and probably at ␣=−0.5 and ␣=−0.8 too兲, indicating the linear independence of the vison states.
Carlo results.6,18 The corresponding energy is ⑀
min= 0.16,
which is significantly lower than the energy 共0.56兲 found above for a completely localized point vison兩i典. The bottom of this variational point-vison band is, however, still high compared to Ivanov’s6 estimate 共0.089兲 of the gap in the
vison sector. The next section provides an improved family of variational states.
D. Dressed vison
The vison wave function of Eq.共39a兲 differs from the RK
wave functions only through minus signs. It has the “acci-dental” property that dimer-dimer correlations are the same as in the ground state.29 As a consequence, the excitation
energy of such a state is carried only by the kinetic part of the Hamiltonian. Clearly, some local reweighting of the dimer configurations in the vicinity of the vortex core would allow an optimized balance between the potential and kinetic costs of the excitation, and would lead to an improved varia-tional wave function. This amounts to “dressing” locally the initial point-vison state by some even—but fluctuating— number of additional point visons.
As a simple improvement of the vison wave function, we introduce a variational parameter␣ to reweight the configu-rations depending on their flippability at the core of the vi-son. This gives the following vison state:
兩i,␣典 =
冑N
1兺
c
共− 1兲Nˆ 共i兲共1 +␣
Fˆi兲兩c典,
Fˆi= Fˆr1共i兲+ Fˆr2共i兲+ Fˆr3共i兲, 共58兲
where Fˆi兩c典=兩c典 if the dimer configuration c is flippable
around one of the three rhombi r1共i兲, r2共i兲, and r3共i兲
contain-ing the triangle i, and Fˆi兩c典=0 otherwise.
As for the point vison of Eq.共39a兲, the vison states of Eq.
共58兲 are not orthogonal and we have to evaluate their
over-laps:
Sij␣=具i,␣兩j,␣典. 共59兲
Repeating the transformation leading to Eq.共48兲, we get
Sij␣=⑀ij具共− 1兲Nˆ 共i,j兲共1 +␣Fˆi兲共1 +␣Fˆj兲典, 共60兲
which is an expectation value for a diagonal operator that we evaluate using Pfaffians. In addition to the sign changes due to共−1兲Nˆ 共i,j兲, some entries of the Kasteleyn matrix have to be modified to incorporate the flippability operators Fˆr. More
precisely, counting only the coverings which satisfy Fˆr= 1 is
done by “isolating” the rhombus r, that is, by switching to zero in the Kasteleyn matrix the 14 bonds which connect the sites of rhombus r to their neighbors outside r.
Beyond some distance between the vison cores i and j 共fourth neighbor on the hexagonal lattice兲, no rhombus can touch simultaneously both triangles. In that case, it can be shown that Sij␣= S0ijis independent of ␣. The eigenvalues of
the overlap matrix S␣共k兲 are displayed in Fig. 8 for a few selected values of␣. Although a full convergence as a func-tion of the truncafunc-tion distance dmax共see caption of Fig.8兲 has
not been obtained, we believe that there is a finite gap for all the values of␣shown here, and that the dressed vison states are linearly independent.
The evaluation of the Hamiltonian matrix elements Hij␣=具i,␣兩H兩j,␣典 共61兲
for dressed vison can still be done using Pfaffians, but the algebraic manipulations are slightly more lengthy than the previous ones, and we refer the reader to Appendix B. We find nonzero matrix elements Hij␣ up to 共and including兲 the
ninth neighbor共compared to first neighbor for point vison兲. The results for the variational dispersion relation are shown Fig. 9. The qualitative shape of the lowest band is almost unchanged compared to the point-vison states 共␣ = 0兲, except for an almost uniform shift which lowers the gap. The minimum of Ek is found at 共kx, ky兲=共/6,/2兲
= B for a variational parameter␣⯝−0.8, and the correspond-ing energy is Emin= 0.119, about 33% higher than the exact
value.
The fact that simple wave functions like those of Eq. 共39a兲 and 共58兲 reproduce qualitatively the shape of the exact
dispersion relation is presumably due to the fact that the dimer-dimer correlation length is rather small at the RK point 共of order of one lattice spacing1兲. As a consequence,
the exact vison states only differ from Eq. 共39a兲 at short
distances from the vortex core, and the long-distance part 共string兲, responsible for the fluxper hexagon, is essentially
FIG. 9. 共Color online兲 Variational vison 关Eq. 共58兲兴 dispersion
relation for different values of␣, along the path A→B→C→A in the Brillouin zone共see Fig.6兲. The curve for␣=0 corresponds to
the variational energies of point-vison states关Eq. 共39a兲兴. The
abso-lute minimum is Emin= 0.119, found at共kx, ky兲=共/6,/2兲=B for a
value⯝−0.8 of the variational parameter␣. The exact value of the gap共0.089兲 共Ref.6兲 is marked as a dotted horizontal line. In this
calculation, the matrix elements Sij␣are neglected for triangles i and j at distance d艌dmax= 10. This value is large enough to ensure a
perfect convergence of the spectrum below Eⱗ0.6, as can be checked from the fact that the dispersion curves obtained with a smaller truncation distance共dmax⯝8.47—shown here with the same colors兲 are practically identical. Some higher energy states, how-ever, are not fully converged.
exact. This is, of course, no longer true away from the RK point and in the direction of the crystal, where the correlation length rapidly grows.
In an attempt to extend this variational approximation away from the RK point, we computed the gap of the dressed vison states in perturbation theory to first order in 共1−V兲. However, at this order, the gap turns out to close very slowly and does not lead to a meaningful estimate for the critical V at the liquid-crystal transition. This failure is closely related to the remark above: the size of the core of the vison pre-sumably grows rapidly away from the RK point, a feature which cannot be accounted for with the present variational states.
IV. CONCLUSIONS
In this paper, we have developed two simple approaches to describe the vison excitations of the QDM on the triangu-lar lattice. The first one is based on a soft-dimer version of the model, which exactly takes the form of a Z2 gauge
theory. We have shown that a semiclassical spin-wave ap-proximation to the spectrum of the dual Ising theory captures the important fact that the disappearance of the RVB liquid is due to a vison condensation.30It also reproduces the
qualita-tive shape of the vison spectrum in the disordered phase and, more importantly, at the transition 共linear spectrum at the correct points of the Brillouin zone兲, as well as the spatial pattern of the ordered crystalline state.
The second approach is a variational approximation to the vison wave functions at the RK point of the 共hard-core兲 QDM. It reproduces semiquantitatively the vison dispersion relation, and provides a simple picture for the vison wave functions which goes beyond the naive point-vison approxi-mation.
Beyond the problem of vison dispersion and condensa-tion, we expect these approaches to be useful for other prob-lems related to vison excitations in QDM, in particular, their mutual interaction and their interaction with vacancies and mobile holes. This is left for future investigations.
ACKNOWLEDGMENTS
We are very grateful to F. Becca, M. Ferrero, D. Ivanov, V. Pasquier, and A. Ralko for numerous insightful discus-sions. F.M. acknowledges the financial support of the Swiss National Fund, of MaNEP, and of the Région Midi-Pyrénées through its Chaire d’Excellence Pierre de Fermat program, and thanks the SPhT 共Saclay兲 and the Laboratoire de Phy-sique Théorique共Toulouse兲 for their hospitality.
APPENDIX A: FOURTH ORDER EFFECTIVE QUANTUM DIMER MODEL
The goal of this appendix is to derive an effective Hamil-tonian for the model of Sec. II A defined by the HamilHamil-tonian
H = HJ+ H⌫= − J
兺
l l x −⌫兺
i兿
l共i兲 l共i兲 z 共A1兲in the limit⌫/JⰆ1 to fourth order in ⌫/J. For simplicity, let us define the unperturbed Hamiltonian H0⬅HJand the
per-turbation V⬅H⌫. Since the Hilbert space of the model is
restricted by the constraint that the number of dimers starting from a given site is odd, the ground state manifold of the unperturbed Hamiltonian for positive J consists of all states having exactly one dimer emanating from each site. Let us denote by P0 the projector onto the Hilbert space generated
by these states, and by S the resolvent defined by S = − 1 − P0
H0− E0
. 共A2兲
It is easy to check that V changes the parity of the number of dimers. Indeed, the term of V acting on the triangle i trans-forms the states with 0 and 3 共1 and 2兲 dimers around the triangle i into each other. This implies that P0VP0= 0 and,
more generally, that the effective Hamiltonian only contains even powers of the perturbation. Thanks to the property P0VP0= 0, the fourth order contribution reduces to three
terms, and the effective Hamiltonian up to fourth order reads Heff= P0VSVP0+ P0VSVSVSVP0− 1 2P0共VS 2VP 0VSV + VSVP0VS2V兲P0.
Up to a constant, this effective Hamiltonian can be written as a QDM acting on four- and six-site plaquettes. The four-site Hamiltonian has the form of the regular RK model with t =⌫2/J−⌫4/J3and V =⌫4/2J3. The six-site Hamiltonian only consists of kinetic terms that flip the dimers around the three possible types of six-site plaquettes shown in Fig. 10, with amplitude −3⌫4/4J3 for types 共1兲 and 共2兲, and with
ampli-tude −⌫4/J3 for type共3兲.
APPENDIX B: HOPPING AMPLITUDE FOR THE DRESSED VISONS
For the dressed vison states关Eq. 共58兲兴, the equivalent of
Eq.共54兲 is Hij␣= 2 N
兺
r0 共c,c¯兲兺
Fr 0共c兲=1 共− 1兲N共c,i兲+N共c,j兲关具c兩共1 +␣Fˆ i兲 +␦i,r0具c¯兩共1 +␣Fˆi兲兴⌸ˆr0关共1 +␣Fˆj兲兩c典 +␦j,r0共1 +␣Fˆj兲兩c¯典兴. 共B1兲 Using the explicit form of the projector⌸ˆr, we getFIG. 10. The three types of six-site plaquettes around which dimer shifts are generated at fourth order in⌫/J.
具i,␣兩⌸ˆr0兩j,␣典 = 1 2N 共c,c¯兲
兺
Fr 0共c兲=1 共− 1兲N共c,i兲+N共c,j兲兵1 +␣ Fi共c兲 −␦i,r0关1 +␣Fi共c¯兲兴其兵1 +␣Fj共c兲 −␦j,r0关1 +␣Fj共c¯兲兴其. 共B2兲Unlike Eq. 共56b兲, both the coverings c and c¯ enter the
expression. This does not have the form of a diagonal observable, and the terms Fr共c¯兲 关with r苸兵r1共i兲,r2共i兲,
r3共i兲,r1共j兲,r2共j兲,r3共j兲其兴 need to be eliminated to allow for an evaluation with Pfaffians. By inspecting the possible relative positions of two rhombi r0and r, one arrives at the following
two relations:
共1兲 If two rhombi r0 and r have zero, one, three, or four
sites in common, Fr共c¯兲=Fr共c兲 for any pair of configurations
共c,c¯兲 which differ by a flip around the rhombus r0.
共2兲 It r0and r have two sites in common, let us call b the
bond of r which does not touch r0. Then we have Fr共c¯兲
= Db共c兲关1−Fr共c兲兴, where Db共c兲=1 if b is occupied by a
dimer of c, and 0 otherwise.
We may combine the two cases above into some compact notation Fr共c¯兲=Ar,r0共c兲, valid for any pair of configurations
共c,c¯兲 which differ by a flip around the rhombus r0.
Accord-ingly, we define an operator Aˆi,r0= Aˆr1共i兲,r0+ Aˆr2共i兲,r0+ Aˆr3共i兲,r0
for each triangle i and rhombus r0. The matrix element of Eq.
共B1兲 is now expressed as the expectation value of a diagonal
operator: Hij␣= ⑀ij 2
兺
r0 具共− 1兲Nˆ 共i,j兲Fˆ r0共1 +␣Fˆi−␦i,r0共1 +␣Ai,r0兲兲共1 +␣F ˆ j −␦j,r0共1 +␣Aj,r0兲兲典. 共B3兲This expression has to be expanded into a polynomial in Fˆ and Dˆ operators before each term can be evaluated thanks to the Pfaffian of an appropriate Kasteleyn matrix. As before the 共−1兲Nˆ 共i,j兲 introduces some sign changes, and each Fˆ 共or Dˆ 兲 operator requires isolating the corresponding rhombus 共or bond兲 by switching to zero the corresponding matrix ele-ments. Several tens of terms typically appear for each pair of triangles共ij兲, and an automated treatment by computer had to be coded to obtain the hopping amplitudes. The results of Fig. 9 represent several hundreds of CPU hours using the softwareMAPLEto generate all the correlators and evaluate the corresponding Pfaffians on a finite lattice with 28⫻28 sites.
*gregoire.misguich@cea.fr
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26See also Refs.11,13, and14for the relation between frustrated
Ising models and theZ2gauge theories appearing in the
descrip-tion of Mott insulators.
27In the kagome lattice共and on any lattice made of corner sharing
triangles, in general兲, this can be done explicitly in a simple way 共Ref.2兲.
28The fact that the Hamiltonian describing the vison hopping is not
translation invariant is due to the fact that the dimer liquid has a nontrivial projective symmetry group共Ref.25兲.
29In the kagome-lattice model of Ref.2, this is an exact eigenstate. 30In the frustrated Ising model formulation of the dimer model,x
measures the presence共−1兲 or absence 共+1兲 of a vison, andz
creates or annihilates a vison关see Eq. 共39b兲, for instance兴. The
classical ground state turns out to have 具z典=0 in the dimer
liquid phase and具z典⫽0 in the crystal phase 共Sec. II B 2兲. So,
the vison creation or annihilation operator acquires a finite ex-pectation value at the transition, the usual signature of a particle condensation. This is almost identical to the standard hard-core-boson↔ spin-12correspondence. If the particle density is repre-sented by 21共1−x兲 共as here for visons兲, magnetic long-range
order in the z共or y兲 direction 共for the spin variables兲 is equiva-lent to Bose condensation共off-diagonal long-range order in the Boson operators兲. An important difference is, however, that the number of visons is not conserved. Only their parity is
con-served. Accordingly, the vison condensed phase spontaneously breaks a discrete共Z2兲 gauge symmetry 共z, which is not gauge
invariant, acquires a finite expectation value in the crystal phase兲, and not a continuous 关U共1兲兴 one. Consequently, the
con-densed phase is gapped and does not have a Goldstone mode. In the dimer model, the spectrum is, therefore, gapless only at the transition.
